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I wonder what are currently the best numerical solvers of ODE for high-accuracy computations. I need an efficient and accurate method to solve ODE that are not pathological (all is smooth) using 128-bit floating-point number (quad arithmetics).

So, what would be good choices (some references comparing different algorithms would be appreciated) ?

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Do you something about the nature of your differential equation? One that involves exponential forms may be very efficiently approximated using exponentials in a way similar to Taylor series...

Utilizing logarithms, xlog(x)... And mixtures of different elementary functions may ask prove useful too... Constructing the method isn't significantly difficult you just need to use regression/basic slope and sum analysis to fit a particular type of function to your curve, rinse and repeat with higher powers.

I would recommend this approach of customizing your own approximation method, especially, if you know something about the nature/behavior of your solution (which you appear to do since you can confidently say its smooth) and want high performance as well as high accuracy.

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  • $\begingroup$ If you can find singularities or know their distribution that too is highly advantageous $\endgroup$ – frogeyedpeas May 15 '13 at 2:42
  • $\begingroup$ It allows you to employ functions of the form 1/(x-a)^s for discontinuities at a, where s is a constant power $\endgroup$ – frogeyedpeas May 15 '13 at 2:43

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